Statistics of Orientation Effects in Linear Polymer Molecules

Abstract

This paper is concerned with the effects of orientation on the combinatorial term g for the number of ways to pack together Nx linear polymers (x mers). Accordingly g is evaluated as a function of the number of molecules in each permitted direction for the case of straight rigid rods. The permitted directions can be continuous so that g is derived as a function of the continuous function f(r) which gives the density of rods lying in the solid angle Δr, or the permitted directions can be discrete so that g is the number of ways to pack molecules onto a lattice. To illustrate the usefulness of the orientation dependent combinatorial terms, liquid crystals are discussed. Another phase is found to exist in addition to the previously predicted nematic phase. This phase is tentatively identified with the cholesteric phase. A procedure is developed for the calculation of the orientation dependent combinatorial term associated with the packing together of molecules of arbitrary shape. A very approximate application of this procedure results in an approximate expression for the combinatorial term which allows one to predict qualitatively the change in the entropy of packing as a function of stretch. It is found that the entropy of packing has the proper behavior to explain the initial deviation of the experimental stress-strain curve from the previous theoretical predictions.

Self-consistent-field modeling of complex molecules with united atom detail in inhomogeneous systems. Cyclic and branched foreign molecules in dimyristoylphosphatidylcholine membranes

“…It was recognized by DiMarzio that parallel bonds located at the same z coordinate cannot block each other. 28,29 This notion enables one to rather accurately calculate the vacancy probability needed for the packing of the chains in the lattice. In effect, an anisotropic field is created that has the property that, when it is large ͑due to the fact that many bonds have a given orientation͒, it will force other bonds to assume this orientation too.…”

A generalisation of the Onsager trial-function approach: describing nematic liquid crystals with an algebraic equation of state

“…In our work we opt for a decoupling approximation of this type, as these have been shown to provide a very accurate representation of the isotropic-nematic transition for hard-core particles of moderate size asymmetry (e.g., [11,42]); further details of the methodology are presented in the following section. Related theories for repulsive lattice models [7,[85][86][87], for models with restricted orientations [88], and for semi-flexible liquid crystalline polymers [89][90][91] are also in common use. It is not possible to mention all of the applications and extensions of the Onsager theory for liquid crystals here; to date of publication there are over two thousand citations to Onsager's 1949 paper since 1970, and the approach is still very popular.…”

Statistical Thermodynamics of Lattice Gas Models of Multisite Adsorption

“…The most well-known analytical approximation is: 1) the theory of Flory-Huggins [48,[55][56][57][58], which is a generalization of the theory of binary solutions of polymer molecules in a monomolecular solvent for the twodimensional case. The fact that in the framework of lattice gas model the problem of k-mer adsorption on homogeneous surfaces is isomorphic to the problem of binary solutions of polymer in a monomolecular solvent, 2) Guggenheim-DiMarzio approximation [59,60], which is based on calculating the number of possible ways of packaging rigid k-mers on lattices with different coordination numbers 3) the approximation based on the extension of the exact solution for a one-dimensional case [52,53] to higher dimensions [49,61], 4) well known quasichemical approximation [62] and mean-field approximation [63], 5) fractional statistical theory (FSTA) for the adsorption of polyatomic molecules, based on Holdan statistics [64], 6) semi-empirical model [61,65], etc. Unfortunately, none of these approximations is universal, and each shows quite good results, depending on the parameters of the model -a flexible or rigid k-mer, the length of k-mer, the presence or absence of lateral interactions between adsorbed molecules, etc.…”